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Von Neumann and Birkhoff ergodic theorems for negatively curved groups. (Théorèmes ergodiques de von Neumann et de Birkhoff sur les groupes à courbure négative.) (English. French summary) Zbl 1359.37007
Summary: We prove maximal inequalities for concentric ball and spherical shell averages on a general Gromov hyperbolic group, in arbitrary probability preserving actions of the group. Under an additional condition, satisfied for example by all groups acting isometrically and properly discontinuously on $$\operatorname{CAT}(-1)$$ spaces, we prove a pointwise ergodic theorem with respect to a sequence of probability measures supported on concentric spherical shells.

##### MSC:
 37A30 Ergodic theorems, spectral theory, Markov operators 22D40 Ergodic theory on groups 20F65 Geometric group theory 20F67 Hyperbolic groups and nonpositively curved groups
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